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Phylogenetic Diversity Measures Based on Hill Numbers Anne Chao National Tsing Hua University Institute of Statistics Hsin-Chu, Taiwan Eco-Stats Symposium The University of New South Wales Sydney, Australia July 11-12, 2012

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Biodiversity Definition Variety and variability among living organisms and the ecological complexes in which they occur Variation of life at all levels of biological organization

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Biodiversity Levels Gene diversity- diversity of genes within a species Species (or taxonomic/phylogenetic) diversity- diversity among species in an ecosystem Ecosystem (or functional) diversity- diversity of different ecosystems on Earth.

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Shannon (1948) entropy Measure of uncertainty uncertainty in the species identity of a randomly sampled individual

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Doubling Property MacArthur (1965), Hill (1973) There are two completely distinct (no overlapped species) communities, each with diversity measure X Combine these two with equal weight, the diversity should become 2X An essential minimum requirement for a “diversity” that ecologists expect “Replication principle” in economics (Dalton 1920): extension to K communities/groups

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“Order” q (Tsallis 2001; Keylock 2005) The order q determines the measure’s sensitivity to species frequencies q > 1, sensitive to common species q < 1, sensitive to rare species q = 1, weighs species by their frequencies, without favoring either common or rare species

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Hill Numbers: “Species Equivalent” “Effective number of species” The number of equally-common species that would be needed to give the same diversity as the community in study For equally-common community, Hill numbers are equal to species richness for all orders of q;

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Phylogenetic Diversity : Community 1 Community 2 All else being equal, which community is more diverse?

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Species in community 2 is more phylogenetically diverse than community 1 Pielou (1975, p. 17) was the first to notice the concept of diversity could be broadened to consider taxonomic difference between species. Community 1 Community 2

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“I think” Tree of Life The first-known sketch by Charles Darwin of an evolutionary tree describing the relationships among groups of organisms s/darwin/idea/treelg.php

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Pioneering work (2) Weitzman (1992, 1993, 1998) from a perspective of economic theory of biodiversity preservation “Unfortunately, Noah’s Ark has a limited capacity …. and a (limited) budget available for biodiversity preservation…” What to preserve?

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The Noah ’ s Ark: the agony of choice The woodpecker might have to go! Courtesy of Ramon Teja,

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Basic approach based on Hill Numbers for shared lineages At any given moment t, slice the tree, we can find the lineage (branch cuts, “species”) and its relative abundance (measure of their importance in the present-day community) Obtain Hill number q D(t) at moment t. Average over from the present time to T years ago Call this average diversity as “Mean Diversity of order q over T years”, it is in units of “lineage” (or “species”).

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Conceptual framework for q = 0 Connect Faith’s PD to mean species richness For a fixed T, the nodes divide the phylogenetic tree into Segment 1, 2 and 3 with duration (length) T1, T2, and T3 In any moment of Segment 1, there are 4 lineages (i.e., 4 branches cut) Segment 2, there are 3 lineages Segment 3, there are 2 lineages The mean lineage (species) richness over the time interval [−T, 0] is (T1/T) ×4 + (T2/T) ×3 + (T3/T) ×2 = total branch length in [-T, 0] / T (Mean Phylogenetic Diversity of order 0 over T years) If T = height of tree, then

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Conceptual framework for q > 0 To incorporating abundance, use lineage abundance: sum of the relative abundances descended from the branch There are T1 assemblages with abundance vector{p1, p2, p3, p4 }, T2 assemblages with abundance vector {p1, p2+p3, p4 } and T3 assemblages with abundance vector {p1+p2+p3, p4 }. There are a total of T1+T2+T3 = T assemblages and each is given the same weight 1/T. The “Mean diversity of order q over T years” is the following average

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Mean Phylogenetic Diversity of order q over T years General Formula B T : all branches in the time interval [-T, 0] L i : the length (duration) of Branch i in the set B T a i : the total relative abundance descended from Branch i

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Related Measure: Branch Diversity q = 0, branch diversity reduces to Faith’s PD Branch diversity: the amount of evolutionary “work” done on the assemblage or the effective lineage- years or lineage-length (or other units) contained in the tree in the time period [−T, 0]

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Traditional Species diversity: Hill numbers for two sites Thinned Site Un-thinned Site

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Order q Site CT (thinned site)Site CU (un-thinned site) q= q= q= Shimatani (2001) concluded that the traditional diversity indices and the taxonomic diversity give different conclusions about the effect of thinning. Our results based on “Mean Phylogenetic Diversity” are consistent with those based on the traditional species diversity for q = 0, 1 and 2.

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Diversity profile Non-phylogenetic: Use a profile of Hill numbers (as a function of order q) to quantify diversity of a community Phylogenetic: Use three profiles (q = 0, 1, 2); each is a function of time T to quantify phylogenetic diversity All these measures satisfy “doubling property”

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Based on species richness (q = 0), the diversity of the thinned site dominates that of un-thinned site for all values of T. But for the common species (q = 1) and very abundant species (q = 2), we have the reverse conclusion. Mean Phylogenetic Diversity Un-thinned Site Thinned Site Un-thinned Site Thinned Site Un-thinned Site Thinned Site